21 PRODUCT OF MEASURES 229 is integrable, by virtue of (13.21.14), hence f®9 is locally v-integrable (13.13.1). Also, for each we Jfc(X) and each v e jJfc(Y), we have [ \f(x)g(y)u(x)v(y) dl(x) du(y) = M. f(x)u(x) d*(x)j ( (g(y)v(y) d^(y}\ by virtue of (1 3.21 .1 4) ; hence the formula (1 3.21 .1 6.1 ) follows from (1 3.21 .1 ). (13.21.17) Let A, IJL be complex measures on X, Y, respectively. Then We may write A =/• |A| and // = g • \u\, where |/| = \g\ = \ (13.16.3). Hence \f®g\ - 1, and the result therefore follows from (13.21.16) and (13.13.4). (13.21.18) Let X be a complex measure on X and in a complex measure on Y. (i) Ifl is concentrated on A cz X and p, is concentrated on B c Y (13.18), then A ® \JL is concentrated on A x B. (ii) Supp(l <g) u) = Supp(/l) x Supp(/x). (iii) With the product convention of (13.11) we have (13.21.18.1) In particular, if X and ILL are bounded, then so is X ® u. By virtue of (1 3.21 .1 7) we may restrict ourselves to the case in which /I and // are positive. Then (i) follows from the fact that X x Y — A x B is the union of the (A ® ^-negligible sets (X - A) x Yand X x (Y - B) (13.21.12). It follows from (i) that Supp(A ® u) c Supp(Jl) x Supp(/i). On the other hand, if x e Supp(A) and y E Supp(/j), then for each compact neighborhood V of x in X and each compact neighborhood W of y in Y, we have A(V)>0 and ^(W) > 0, whence by (13.21.15) (A®M)(VxW) = l(V)/i(W) > 0. This establishes (ii), by virtue of the definition of neighborhoods in X x Y. Finally, (iii) follows from (13.21.11) applied to/= (px and# = <pY, except when one of the factors on the right-hand side is 0 and the other is +00; but in this case we have A® ^ = 0 by (13.21.12), and therefore the formula (13.21.18.1) remains valid in this case. (13.21.19) There are analogous definitions and results for the product of any finite number of measures : if (X^ ^ ^ „ is any finite sequence of locally compactgligible